Biomedical Engineering Reference
In-Depth Information
8.5.1 Tensor Invariants
Tensor invariants (rotational invariants) are combinations of tensor elements
that do not change after the rotation of the tensor's frame of reference, and
thus do not depend on the orientation of the patient with respect to the scanner
when performing DT imaging. The well-known invariants are the eigenvalues
of the diffusion tensor (matrix)
d
, which are the roots of the corresponding
characteristic equation
3
2
−
C
1
·
λ
+
C
2
·
λ
−
C
3
=
0
,
(8.27)
λ
with coefficients
C
1
=
D
xx
+
D
yy
+
D
zz
C
2
=
D
xx
D
yy
−
D
xy
D
yx
+
D
xx
D
zz
−
D
xz
D
zx
+
D
yy
D
zz
−
D
yz
D
zy
(8.28)
C
3
=
D
xx
(
D
yy
D
zz
−
D
zy
D
yz
)
−
D
xy
(
D
yx
D
zz
−
D
zx
D
yz
)
+
D
xz
(
D
yx
D
zy
−
D
zx
D
yy
)
.
Since the roots of Eq. (8.27) are rotational invariants, the coefficients
C
1
,
C
2
,
and
C
3
are also invariant. In the eigen-frame of reference they can be easily
expressed through the eigenvalues
C
1
=
λ
1
+
λ
2
+
λ
3
C
2
=
λ
1
λ
2
+
λ
1
λ
3
+
λ
2
λ
3
(8.29)
C
3
=
λ
1
λ
2
λ
3
and are proportional to the sum of the radii, surface area, and the volume of the
“diffusion” ellipsoid. Then instead of using (
λ
1
,λ
2
,λ
3
) to describe the dataset,
we can use (
C
1
,
C
2
,
C
3
). Moreover, since
C
i
are the coefficients of the character-
istic equation, they are less sensitive to noise than are the roots
λ
i
of the same
equation.
Any combination of the above invariants is, in turn, an invariant. We consider
the following dimensionless combination:
C
1
C
2
/
C
3
. In the eigenvector frame of
reference, it becomes
C
1
C
2
C
3
=
3
+
λ
2
+
λ
3
λ
1
+
λ
1
+
λ
3
λ
2
+
λ
1
+
λ
2
(8.30)
λ
3
and we can define a new dimensionless anisotropy measure
C
1
C
2
C
3
−
3
1
6
C
a
=
.
(8.31)
